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# test3-05 - g using the information you obtained in parts(a...

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Department of Mathematics, University of Toronto Term Test 3 – March 15, 2006 MAT 137Y, Calculus! Time Alloted: 1 hour 50 minutes Examiners: I. Alexandrova, M. Harada, V. Ivrii, G. Lynch, M. Saprykina, A. Savage 1. Evaluate the following integrals. (8%) (i) Z dx x ln x . (8%) (ii) Z x 2 arctan xdx . (10%) (iii) Z x 3 - 4 x - 1 x ( x - 1 ) 3 dx . (10%) (iv) Z dx ( 9 - x 2 ) 5 / 2 . 2. Let g ( x ) = R x 0 f ( t ) dt , where f is the function whose graph is shown below. (4%) (a) Find the values of x where the local maximum values of g occur, and ﬁnd the values of x where the local minimum values of g occur. (4%) (b) Where does g obtain its absolute maximum value? Justify your answer. (4%) (c) On what intervals is g concave upward and concave downward? (4%) (d) Sketch the graph of

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Unformatted text preview: g using the information you obtained in parts (a) thru (c). (12%) 3. Find the area of the region bounded by the parabola y = x 2 , the tangent line to the parabola at the point ( 3 , 9 ) , and the x-axis. (12%) 4. Find the volume of the solid that is generated by rotating the region bounded by the curves y = x 2 , y = 8-x 2 about the line y =-1. 1 5. (8%) (i) Find the absolute maximum and absolute minimum values of the function f ( x ) = xe-x on the interval [ , 2 ] . (8%) (ii) Evaluate lim x → + ( 1 + sin5 x ) cot x . (10%) 6. Prove for all x > 0, x-1 x ≤ ln x ≤ x-1 . 2...
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test3-05 - g using the information you obtained in parts(a...

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